Convexity issues in multivariate multiple testing of treatments vs. control

Abstract

The problem of multiple testing of each of several treatment mean vectors versus a control mean vector is considered. Both one-sided and two-sided alternatives are treated. It is shown that typical choices for marginal test procedures will lead to step-down procedures that do not have convex acceptance regions. This lack of convexity has both intuitive and theoretical disadvantages. The only exception being linear tests in the one-sided problem. Although such a procedure is atypical, it not only has convex acceptance regions but is such that critical values are obtainable so that the overall procedure can control FDR or FWER. For both one-sided and two-sided alternatives, two other stepwise multiple testing methods are presented that do have convex acceptance regions.